We characterize the (1, 1) knots in the three-sphere and lens spaces that
admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these
manifolds admit non- trivial L-space surgeries. We also recover ...

The main functions of the kidney take place in the nephrons. For their proper operation,
nephrons need to be supplied with a stable blood flow that remains constant despite
fluctuations of arterial pressure. Such stability ...

Noise-induced stabilization occurs when an unstable deterministic system is stabilized
by the addition of white noise. Proving that this phenomenon occurs for a particular
system is often manifested through the construction ...

Many numerical approaches exist to solving models of electrical activity in the heart.
These models consist of a system of stiff nonlinear ordinary differential equations
for the voltage and other variables governing channels, ...

This thesis studies properties of edge toric ideals and resolutions by analyzing the
associated graphs of algebraic structures. It mainly focused on proving that the repeated
edges in a graph wouldn't change some properties ...

We here developed a third-order accurate numerical method for scattering of 3D electromagnetic
waves by doubly periodic structures. The method is an intuitively simple numerical
scheme based on a boundary integral formulation. ...

To study periods of fundamental groups of algebraic varieties, one requires an explicit
algebraic de Rham theory for completions of fundamental groups. This thesis develops
such a theory in two cases. In the first case, ...

This thesis is concerned with a structure called the Reeb graph. There are three main
problems considered. The first is devising an efficient algorithm for comnstructing
the Reeb graph of a simplicial complex with respect ...

The central idea of this dissertation is to interpret certain invariants constructed
from Laplace spectral data on a compact Riemannian manifold as regularized integrals
of closed differential forms on the space of Riemannian ...

This dissertation places intersection homology and local homology within the framework
of persistence, which was originally developed for ordinary homology by Edelsbrunner,
Letscher, and Zomorodian. The eventual goal, begun ...

This dissertation extends the theory of persistent homology to time varying systems.
Most of the previous work has been dedicated to using this powerful tool in topological
data analysis to study static point clouds. In ...

Interacting particle systems have been applied to model the spread of infectious diseases
and opinions, interactions between competing species, and evolution of forest landscapes.
In this thesis, we study three spatial models ...

This work introduces geometric and topological data analysis (TDA) tools that can
be used in conjunction with sliding window transformations, also known as delay-embeddings,
for discovering structure in time series and dynamical ...

We develop algorithms to approximately count perfect matchings in bipartite graphs
(or permanents of the corresponding adjacency matrices), perfect matchings in nonbipartite
graphs (or hafnians), and general matchings in ...

This thesis analyzes the Chow motives of 3 types of smooth projective varieties: the
desingularized elliptic self fiber product, the Fano surface of lines on a cubic threefold
and an ample hypersurface of an Abelian variety. ...

We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau
threefolds that are of the form $\CC^3/\ZZ_r$, focusing on the cases where the orbifold
has an isolated singularity. We prove a lower bound ...

When simulating multiscale stochastic differential equations (SDEs) in high-dimensions,
separation of timescales and high-dimensionality can make simulations expensive. The
computational cost is dictated by microscale properties ...

To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation
category.An exact Lagrangian cobordism between two Legendrian knots gives a functor
of the augmentation categories of the two knots.We study ...